Download Study Guide and Intervention Parallelograms

NAME
DATE
6 -2
PERIOD
Study Guide and Intervention
Parallelograms
Sides and Angles of Parallelograms
A quadrilateral with
both pairs of opposite sides parallel is a parallelogram. Here are four
important properties of parallelograms.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
S
Q
R
If PQRS is a parallelogram, then
−−− −−
−− −−−
PQ SR and PS QR
If a quadrilateral is a parallelogram,
then its opposite angles are congruent.
∠P ∠R and ∠S ∠Q
If a quadrilateral is a parallelogram,
then its consecutive angles are
supplementary.
∠P and ∠S are supplementary; ∠S and ∠R are supplementary;
∠R and ∠Q are supplementary; ∠Q and ∠P are supplementary.
If a parallelogram has one right angle,
then it has four right angles.
If m∠P = 90, then m∠Q = 90, m∠R = 90, and m∠S = 90.
Example
If ABCD is a parallelogram, find the value of each variable.
−−
−−−
−− −−−
AB and CD are opposite sides, so AB CD.
2a
A 8b°
B
2a = 34
a = 17
112°
∠A and ∠C are opposite angles, so ∠A ∠C.
D
C
34
8b = 112
b = 14
Exercises
Find the value of each variable.
1.
8y
2.
3x°
6x°
4y°
88
x = 30; y = 22.5
3.
6x
3y
x = 15; y = 11
4.
6x°
12
x = 2; y = 4
5.
55°
60°
5x°
12x°
x = 10; y = 40
2y
6.
30x
2y °
150
72x
x = 13; y = 32.5
Chapter 6
3y°
x = 5; y = 180
11
Glencoe Geometry
Lesson 6-2
If a quadrilateral is a parallelogram,
then its opposite sides are congruent.
P
NAME
DATE
6 -2
Study Guide and Intervention
PERIOD
(continued)
Parallelograms
A
Diagonals of Parallelograms
Two important properties of
parallelograms deal with their diagonals.
B
P
D
C
If ABCD is a parallelogram, then
If a quadrilateral is a parallelogram, then
its diagonals bisect each other.
AP = PC and DP = PB
If a quadrilateral is a parallelogram, then each
diagonal separates the parallelogram
into two congruent triangles.
ACD CAB and ADB CBD
Example
A
Find the value of x and y in parallelogram ABCD.
6x
The diagonals bisect each other, so AE = CE and DE = BE.
6x = 24
4y = 18
x=4
y = 4.5
B
18
E 24
4y
D
C
Exercises
Find the value of each variable.
1.
2.
4y
3x
12
8
28
4.
10
30°
y
60°
4x
x = 7; y = 14
5.
4y°
2x°
12
3x°
x = 15; y = 7.5
2y
6.
4
x
17
3x
1
x = 3−
3
; y = 10 √
3
y
x = 15; y = 6 √
2
x = 15; y = √
241
COORDINATE GEOMETRY Find the coordinates of the intersection of the
diagonals of ABCD with the given vertices.
7. A(3, 6), B(5, 8), C(3, −2), and D(1, −4)
8. A(−4, 3), B(2, 3), C(−1, −2), and D(−7, −2)
(3, 2)
(-2.5, 0.5)
A
9. PROOF Write a paragraph proof of the following.
Given: ABCD
Prove: AED BEC
B
E
D
C
−− −−
−− −−
Diagonals of a parallelogram bisect each other, so AE CE −−
and BE
−− DE .
Opposite sides of a parallelogram are congruent, therefore AD BC.
Because corresponding parts of the two triangles are congruent, the
triangles are congruent by SSS.
Chapter 6
12
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
x = 4; y = 2
3.
2y